Mechanical response of polyether polyurethane foams under multiaxial stress and the initial yielding of ultrathin films

摘要

In the first part of this thesis, we study the mechanical response of elastic polyether polyurethane (EPP)foams by means of experiments, theory, and modeling. The experiments include five loading cases: uniaxialcompression along the rise direction; uniaxial compression along two mutually perpendicular transversedirections; uniaxial tension along the rise direction; shear combined with compression along the rise direction;and hydrostatic pressure combined with compression along the rise direction. We use a commercial series offive EPP foams of apparent densities (mass per unit volume of foam) 50.3, 63.0, 77.0, 162.9 and 220:5 kg/m3.We perform a test for each foam in the series and each loading case. In every test we measure the mechanicalresponse in the form of a stress-strain curve or a force-displacement curve; in several tests we use a DigitalImage Correlation (DIC) technique to compute the strain fields on the surface of the specimen.For some loading cases, including uniaxial compression along the rise direction, the mechanical response ofthe three foams of lower density exhibits a stress plateau. This stress plateau has been commonly interpretedas a manifestation of a bifurcation of equilibrium (Euler buckling of the microstruture of the foam), a globalphenomenon that encompasses the entire microstructure of the foam at once. In this interpretation, theplateau stress (i.e., the value of stress on the stress plateau) is the eigenvalue associated with the bifurcationof equilibrium. Nevertheless, our experimental results indicate that a stress plateau is invariably accompaniedby heterogeneous, two-phase strain fields, consistent with the occurrence of a configurational phase transition.Thus we argue that the plateau stress is the Maxwell stress associated with the attainment of a limit point(snap-through buckling of a cell of the foam), a local phenomenon which progressively sweeps through themicrostructure of the foam.For other loading cases, including uniaxial compression along a transverse direction, the mechanicalresponse does not exhibit a stress plateau, and the stress-strain curves harden monotonically regardless ofthe density of the foam. The strain fields remain homogeneous, even for the least dense foam.We use our experimental results to calibrate a mean-field model of EPP foams. In this model, a unitcell composed of several bars is cut off from an idealized, perfectly periodic foam microstrusture. The tipsof the bars of the cell are subjected to a set of displacements affine with the applied mean deformationgradient, and left to rotate freely. The unit cell is characterized using a few physically meaningful materialand geometric parameters whose values may be readily estimated for any given foam.We verify that under uniaxial loading the model predicts configurational phase transitions, stress plateaus,and two-phase fields for low-density foams; a critical point for foams of a critical density; and monotonicallyhardening stress-strain curves for foams of density higher than the critical density. The critical exponentsassociated with the critical point are the same as in other mean-field models such as the Van der Wallsmodel of a fluid.With a suitable choice of parameters, the model gives predictions that compare favorably with ourexperimental results for all loading cases. In particular, the model gives a nonconvex strain energy functionwhere (and only where) the experiments exhibit a stress plateau and two-phase strain fields.We conclude that the mechanical response of EPP foams is dominated at large strains by either one oftwo mechanisms at the level of a foam cell: snap-through buckling, which leads to nonconvex strain energyfunctions, stress plateaus, and two-phase strain fields; or bending, which leads to convex strain energyfunctions, monotonically increasing stresses, and homogeneous strain fields.This conclusion allows us to interpret an extensive series of experiments in which EPP foam specimens arepenetrated with a wedge-shaped punch. For low-density foams, we find experimentally that the mechanicalresponse is linear up to a penetration of the punch of about 40% of the height of the specimen. We surmisethat the strain field in the specimen consists of a high-strain phase in a region close to the tip, where a phasetransition has taken place, and a low-strain phase in a region far from the tip, where the phase transition isyet to take place. The two regions are separated by a sharp interface, where the strain is discontinuous. Weuse DIC to trace the sharp interface as it grows and sweeps through the specimen during a test.By studying theoretically the self-similar growth of a sharp interface, we predict a linear response withinthe self-similar regime, in accord with our experimental findings. We then apply the same theory to the caseof a conical punch, predict a quadratic response within the self-similar regime, and verify our prediction byperforming experiments with a conical punch. We conclude that in the self-similar regime the mechanicalresponse is ruled entirely by geometry and depends only on the dimensionality of the punch and the plateaustress of the low-density foam.In the second part of this thesis, we study the initial yielding of ultrathin metallic films (thickness of afraction of a micrometer). Recent experiments indicate that in free-standing metallic films of constant grain sizethe initial yield stress increases as the film becomes thinner, it peaks for a thickness on the order of 100 nm,and then starts to decrease. This reversing (first hardening, then softening) size effect poses two challenges:(1) It cannot be explained using currently available models and (2) it appears to contradict the little-knownbut remarkable experimental results of J. W. Beams [1959], in which the size effect in bulge tests did notreverse even for a thickness of 20 nm.We show that the reversing size effect can be explained and the contradiction dispelled by taking intoaccount the effect of the surface stress on the initial yielding. We also predict that the mode of failure ofa film changes from ductile to brittle for a thickness on the order of 100 nm, in accord with experiments.Our successful application of methods of continuum mechanics to films as thin as 100 times a typical latticeparameter adds to a growing realization of the robustness of these methods at ultrasmall length scales.